Population balances in case of crossing characteristic curves: Application to T-cells immune response

TL;DR

This paper introduces a new Transport Method to accurately model cell population dynamics with crossing characteristic curves, specifically applied to T-cell immune responses involving complex protein interactions.

Contribution

A novel Transport Method is developed to handle crossing characteristic curves in population balance equations, improving solution smoothness at crossing points.

Findings

The method successfully models T-cell population dynamics with crossing characteristics.

It provides smooth solutions where classical methods fail.

Applicable to complex biological systems with interacting variables.

Abstract

The progression of a cell population where each individual is characterized by the value of an internal variable varying with time (e.g. size, weight, and protein concentration) is typically modeled by a Population Balance Equation, a first order linear hyperbolic partial differential equation. The characteristics described by internal variables usually vary monotonically with the passage of time. A particular difficulty appears when the characteristic curves exhibit different slopes from each other and therefore cross each other at certain times. In particular such crossing phenomenon occurs during T-cells immune response when the concentrations of protein expressions depend upon each other and also when some global protein (e.g. Interleukin signals) is also involved which is shared by all T-cells. At these crossing points, the linear advection equation is not possible by using the classical way of hyperbolic conservation laws. Therefore, a new Transport Method is introduced in this article which allowed us to find the population density function for such processes. The newly developed Transport method (TM) is shown to work in the case of crossing and to provide a smooth solution at the crossing points in contrast to the classical PDF techniques.